NATIONALADVISORYCOMMI’M’EEFORAERONAUTICS

TECHNICALNOTE2918

EFFECTS OF PARALLEL-JET MIXING ON DOWNSTREAM MACH

NUMBER AND STAGNATION PRESSURE WITH APPLICATION

TO ENGINE TESTING ~ SUPERSONIC TUNNELS

By HarryBernstein

LewisFlightPropulsionLaboratoryClevelamd,Ohio

Washington

March 1953

7%(?

.

---- . . . . . .. . . . . . . . . .. . . ... . .. . . .. .. .....— ——. . . .. .

TECHLIBRAHYKAFB,NM,

lR MATIOIWU2ADVT30RYCC3MDZR3FORAEROIUUJTIC““--

EEL?ECTSOFPf&WzEL—JETMIXINGOIVDCWJ’JSTRWWMAOHJ!WMKER

IIOLLIID5

Am Em-m PRmsclmWITHAJ?PIW3.ATION

TESTINGINSUPERSONICTUNNELS

BYIWTY Bernstein

SUMMARY

TOENGINE -

A one-dimensionalanalysisof theresultsoftheparallel-j~mixingencounteredtithetestdngofengtiesin supersonictidtunnelsisreported.Eqzatiauswerederivedfordetermin~thetotalpressureandMachnumberbehtidthetunnelterdnalshock.Themethodrepresentsa simpleprocedurefordetexminingthesequantitieswhilea tunnelisstillinthedesi~ stage.A specificexampleofthemethodistacluded.

.

Theresultsofpressures,andMachtestingofengines.withthetunnelair.

nl’TRoDumoN

mfxingtwostreamsofd,iffermttemperatures~numbersareofimportanceinthewind-tunnelIna tunneltheexhau6tgasesoftheengtiemixTheeffectof sucha mixinguponthetotal

pressureandMch numberofthecombinedstreamsisofparticulartiterest.Theevaluationoftheseeffectsas functionsofen@ne-perfonanceparametersisof impo~oe indeterminingthetunnelpowerrequirements.

Thepresenttivestigatim,madeattheMCA Lewislaboratory,isconcernedtiththeanalysisoftheresultsofthismixingprocessbymeansofone-dimensional-fluwequations.Themalysisisrestrictedtothecaseinwhichtheareaoftheresultmtstreamisequalto thesumoftheareasoftheorig~l streans.As a result.ofthisrestricttm,thesolutimismadedependautupontheupstreamconditimsonly.TM existenoeofanyareachangeswouldmakeitnecesseryto evaluateanyaxialforcesW ticludetheminthemomantumequation.Suchforceswouldbe dependentuponthestrengthsandpositionsofmy shocksinthemixlq regicm;theseinturnwouldbe dependentuponthedownstreamstaticpressure.

.- — .— —.— --

2 N.ACATN 2918

Thesoluticm,when-a changesexist,isthereforedependent,foragivensetofupstreamconditions,uponthedownstreamstaticpressure.

Thesoluticmaspresentedisalsoapplicableto problemsassociatedwiththemtzinginlongcylindricalejectorsandto anyotherproblems

,,N

towhichtherestrictionsmadeintheanalysisapply.0)P

A

CP

H

K

M

43

%f

%

P

P

T

t

x

Y

a

Y

STM1301S

Thefollowingsymbolsareusedinthisreport:

streamcross-sec-tionalarea

specifioheatofatiat constm.tpressure

heatingvalueoffuel,13tu/13

Q3/Ql,deftied~ eq.(9)

Machnumber

airflowthroughenghejlb/see

fuelmass-fluwrate,lb/seo

totalmass-flowrateoftwostreamsbehg mixed,lb/see(massoffuelnotticluded)

totalpressure

staticpressure

totaltemperature,%

statictemperate,%

definedh eg..(7)

definedineq.(8)

productofareamd staticpressureratiosoftwostreamsbe~mixed,A2P2/AIPl

ratioof specifioheatsforair

-1

.

-— -.

NACATN 2918

,

ATacrosscombustimchamber7C combustionefficiencyofengine, maximumpossibleAT

e ratioof

E ratioof

‘c ratioof

statictemperaturesinstreamsbeingmixed,t2/tl

Maohnumbersof stre- beingmixed,M2/M1

totaltemperaturesin streamsbeingmixed,T2/T~

9 a functionofMmh number, \ J

(1+ ;2)2

Subscripts:

m measuredvalue(fig.6)

max JMXXimUmmeasuredvalue(fig.6)

o conditions

1 conditions

2 conditicms

3 Conaitims

in

in

in

engineinletstreamtube

tunnelstreamadjacenttoengineexitstation

exhaudstreamat engineexitstaticm

aftermixing

behindnormalshook(correspondingto nnprimedconditions

Superscripts:

1 conditionsof samesubscriptaheadof shook)

It conditionsinresultantstreamafterheatadditioninmixingregion(correspond@to qumtitiesof samesubscriptbeforeheatadditian)

.

MIXtNGEQUATIOIiS

Derivation

Theanalysisisrestrictedtothecaseof one-dimensicmalflow,wheretheareaofthestreamsafterm- isequaltothesumoftheareasofthetwostreamsbehg mixed(seesketch}.Themomentumequatianmaycontainno bodyforcetermsbetweentheinitialandfinalstationsatw.blchthefluwisuniform.

——— — -———

4 NACATN 2918

Mixhg region

.1 I2

Theequaticmsrelatingccmditions“lefore‘andafterthemtxlngprocess=e (sticeA1+A2=A3):

Momentumequaticm

(1)PIA@ + ?&) + P#2(l + 7M22)= P3A3(1+ 7M32)

Energyequation

( ) (-1

)P#-&~ 1++~2 +PZ$2*M21+ +%2

J(-17

)= p3A3 t3M3 1 + 2 M32

Continuityequaticm

Theassumptionavoidthenecessity

(2).

ofof

PIAIMI P##2 P3A3M3—=*’6 &

(3)

cmstant 7 and ~ wabmadeinordertoa trial-ad-errwsolution.Thevalidityof

thisassumptionisdiscussedinappendbA.

Theeffectsofviscosityh theboundarylayeralongthewallofthechannel(frictionforce)areneglectedthroughoutthismlysis.

— ——. —.—— —— —

J!?ACATN 2918 5

h mosttunnelssuitablerelativetothepressureTheeffectsofviscosityirrelevzmtifcmditicm

xl’Dr Divisionofeachof

termfields

for

and

enginetesting,thisforceissmall.mmentulac-es ofthemainstream.turbulenceh themixingprocessare

at statim3 =e uniform,asassumed.

equaticms(1),(2),and (3)by itsftist

. where a,c, and 6’aredeftiedas

r

TheparametersXand Y~e) .

(4)

(5)

(6)

definedas follows:

1 + w%‘

Equations(4),(5),and (6)mayhe combtiedto eltite theU@KWllS p3 and t3, leav@ M3 asthemly remainingunlnown.Multiplyingequations(5)and (6),divid

equation(4),andsetting,2(1 ,+qb:;:; ‘f

(1+.@)2

( J(1+ OXY+3)1+=~3e

(1+M)2 ‘m==. whichshsllbe designatedthemixingequation.

(7)

(8)

(9)

—— — .

6

Theknowncanditicznsat statians1to findthevalueof Q3. Thefunctim

lJACATN 2918

md 2 andequation(9)areusedg iSplotteda@n8t Mach

numberinfigure1. Thecurvesshowthatforvaluesof q greaterthmapproximately0.105,M isdouble-valuedin Q. Thetwovaluesof MarethesupersmicandsubsonioMachnumbersoneithersideofa normalshock.Thismayhe shownanalyticallyfroma solutionoftheequatimsgovezmjngchangesacrossa nozmalshock.Asno restrictionsweremadeinthederivaticm,itis3nferredthat(forthesubsonicsolution)thefhal cmditimsareindependentoftheorderof occurrenceoftheprocessesofm~n andshock.E additimjtheexistenceofonly

.

twosoluticmfortheMachntier tier mixing@lies tidependenceofthetypeof shocksystmntithemixingregion.

To determinewhichvalueof M3m he distinguished:

(a)Themixingstreamsarebothsolutim M3t isalwaysthecorrect

isphysicallycorrect,threeoases

subsonic.h thiseasethesubsonioone,sincethesuperscmiosolution u

tillresulttia netdecreaseinentropy.

(b)Them&ing streamsarebothsupersonic.kthem~of twosupersonicstreamsbothsolutionsarephysicallypossible.Thesuper-sonicsolutionwilloccurwhenthecunditimsrequirethatthebackpressure(staticpressureaftermmklnn)be low. J3?thisbackpressureishigh,shockswillexistinthemixhg regim andtheresultitstreamwXU.be sulsonic.Choiceofthesolutionisthereforedictatedby theparticularproblemtowhichtheresultsareapplied.

(c)Onestreami.ssupersonicwhiletheotherissubsonio.Inthiscasethesubsmicsoluticmisalwayspossible.Thesupersonicsolutionisvalidh caseswheretlmsubsonicjetismch smallerthahthesupersonicstream.Theexistenceof sucha supersonicsolutimisdetenntiedby an investigationoftheentropyohemgeintheprocess.H bothsolutions=e possible,thechoiceisagatidepandentupmtheparticularproblemunderconsideration.

temperature-(Z= 1)reduoesto

Whena solutim(eitherfromeq.(9)and(6)to find P3

willproblemofmixingtwostreamsofequaltotalarise.Here,since e =z/Y, equatim(9)

(lo)

fortheMachnumberaftermixinghasleanobtained●

2or eq..(10)),substitutionmaylemadeInequations(4)and t3,respectively. .

—— —. . — —

NACATN 2918 7

Discussion

NJ”e‘1+

r

.

Thecomplexityofthemixingequationmakesitdifficultto determine,by inspection,theeffectsupontheresultantstreamcausedby variationinthevaluesofeachofthenondimensionalparameters.Inan efforttoillustratescnneoftheseeffects,samplecalculationsforan Ml of3.0wereperfomed.Theresultsareplottedin figures2 and3. ThesubsonicJ@chnumberaftermixingM31 isplottedinfigure2 asafunctionoftheparametera fora fewchosencombinationsof ~ ande. Thetotal-pressureratio P31/plisplottedinfigure3 asa functionofthesameindependentvariables.Thecombinationsof ~ and 8 werechosento illustrateseparatelytheeffectsofdifferentMachnumbersanddifferentstatictemperaturesinthemixingstreams.

As seeninfigure2,mixingstreamsofdifferenttemperatures(13# 1)causedan increaseinthesubsonicMachnumleraftermixtigoverthereferencevalue (~= 1, 0 = 1;no?mal-shocksolutionforMl = 3.0). Thisincreaseisgreatestat a = 1,foranyvalueof 6.As a approachesO or ~, the M3t curvesapproachthereferencelineasymptoticallyaswouldbe expected,sincethesevaluesof a implythattheareaofoneofthemixingstreamsisnegligiblewithrespecttptheother.

TheeffectsofmixingstreamsofdifferentMachnumber (E+ 1)me observedto varygreatlywiththevalueof ~; however,a fewgeneralconclusionsaboutthesevariationsmaybe stated.Foranyvalueof ~, thecurveisasymptotictothereferencelineas a approacheszero.If a valueof g > 1 isused,theresultingcurverisestoapeakat somevalueof a lessthanunity.Thecurvethendropsbelowthereferencelineandapproachesa valueof M31 asymptoticallyasa approaches=, Thisvalueof M31 isthesubsonicMachnumberbehinda normalshockata MachnumberM2 (M2= EM1). For ~e 1,thecurvesriseabovethereferencelineandreacha peakat svalueof a largerthanthosevaluesshownon thefigure.(Differ-entiationofthemixingequaticmwithrespectto a showsthatthecurveofresultantMachnumberhasonepeakforanyvaluesof ~ ande,exceptthecombinationK = 1, 13= 1.) As a approaches- thesecurvesarealsoasymptoticto someMachnumberM3~. ThisvalueiseitherthesubsonicMachnumberbehinda shockatMachnumberM2(for‘M2> 1),or M2 itself(forM2< 1).

b thecomputationsforthetotal-pressureratiosP31/pl,someassmptionhadtobemadeconcerningthevariablea. WhiletheMachnumbercurves(fig.2)arequitegeneral,thetotal-pressure-ratio

8 NACATN 2918

curvesdependuponthe‘actualcomponentsof a, t~t is,upon P2/P1 adA2/Al (seeeq.(4)). Theratio p2/plwasthereforearbitrarilychosentobe unity,andhence a,infigure3,representsthearearatioA2/Al.

For < = 1 anddifferentstatictemperatures(f3~ 1),thepressure-ratiocurves(fig.3(a))fallbelowthereferenceline,becomeasymptotictothislineas a approachesO or m, andhavea mintiumpointata= 1. Thedecreaseintotalpressuroisdueto theheatexchangebetweenthestreamsbeingmixedandthenetentropyincreaseassociatedwiththisheatexchange.

For g+ 1 (unequalMachnumbers),thecurves(fig.3(b))areasymptotictothereferencel.lneas a approacheszero.As a increases,thecurvesdepartfrmnthereferenceline,fallingaboveorbelowitas5 isgreaterthanorlessthanunity,respectively.As a approachesinf=ty, thesecurvesapproachthevaluesJ?2,/Pl(forM2 =Wl> 1)or P2/Pl (forM2< 1).

.

APPIZCA’ITON‘IOECWCNETESTIITGINSUPERSONICWINDTUNNELS

Theequationsjustderivedmaybe appliedinthedesignofa super-sonicwindtunnelinwhichenginesaretobe tested.In sucha tunneltheexhaustgasesoftheen@e mixwiththetunnelair;thism~gaffectsthevaluesoftheresultantMachnumberandtotalpressure.Theevaluationoftheresultantstreampropertiesas functionsofengine-perfomnanceparametersisoftiportanceindeterminingthepowerandpressure-ratiorequirementsofthetunnel.Thearrangementisschematicallyillustratedinfigure4.

At theengineexitstation,meanvaluesoftheflowpropertiesofthetunnelstream(station1)arerequiredfortheone-dimensionalanalysis.If A2/Ao is closetounityandtheengineis operatingatamass-flowratioofunity,thesemeanconditionsmaybe takenasfree-streamconditions.Moreaccuratemeanvaluesmaybe obtained,ifnecessery,by constructingtheflowfieldpasttheoutersurfaceoftheengine.

I?lowpropertiesh theexhaustjetattheengineexit(station2)maybe evaluatedas functionsofengine-performanceparameters(%/%,

—

M2,‘c).Again,theassumptionofan engineoperatingatamass-flowratioofunityismade. Ifthemassofthefuelisneglected,theconditionof constantmassflowyieldsthefollowingrelationbetween .enginepressurerecovery,exhaust-jetMachnumber,andtotal-temperatureratio: .

2R

.

.

NACATN 2918

=%J!!]+(,1)If thegeometryoftheenginetobe tested(M2 isa functionofexhaust-nozzlegeometryonly)and T areknown,equation(11)profidesa meansof calculatingthetotalpressureintheexhaustjet.

Oncetheconditionsintheexhaustjetandtheadjacenttunnelstreamhavebeendetermined,usemaybe madeofthemixingequation(eq.(9))to evaluatetunnelMachnumberandtotalpressureaftermixing.Forthisapplication,station3 mustbe assumedtobe a suitabledistancedownstreamofthetunnelterminalshock,=d hence,thesubsonicsolutionto themixingequationisthepropersolution.Thereasonsforthisassumptionareclearlyillustratedinfigures5 and6. lHgure5showsa jetexhaustingintoa supersonicstream.Thejetis olservedto expandslightlyto satisfyambientstatic-pressureconditions,butlittlemixingisobserved.Thisisfurtherevidencedby the‘temperatureprofilespresentedinfigure6. Theseprofiles,whichareaffectedhytheamountofmixing,showthatthemajorportionofthemixingoccursina regiondownstreamofthetumnelterminalshock,andhencetherequirementthatstation3 be a suitabledistancedownstreamoftheterminalshock.

AdditionalchangesinMachnumberandtotalpressuremaybe causedby heatadditicminthemixingregion.Thisisusuallythecaseinenginetesting,forexcessfuelis carriedoutoftheengineintheexhaustjet. Thisfuellmrnsinthemixingregionjustdownstreamofthetunnelterminalshock.Figure7 illustratesthiseffect.Inamnuchas littlefuelisusuallycarriedoutwiththee-ust @s~ as c~p~dwiththetotalmass-flowrate %, the~~itude of~Y c~ges dueto thisburningmaybe small,andinmanycasescanbeneglected.JhappendixB,relationsarederivedwhichmaybe usedinevaluatingthechangesduetothisheataddition,ifa higherdegreeofaccuracyisdesired.

Throughoutthisanalysisthetunneltermtialshockwasassumedtobe locatedinthetestsectiondownstreamofthemodel.Thisrepresentspeakefficiencyoperationofa tunnelhavingno secondthroat.Whilethisispossible,mosttid tunnelsarenotoperatedat peakefficiencybutareoperatedwiththeterminalshockpositioneda shortdistancedownstreamofthestartofthetunneldiffuser.Ifthisisthecase,theresultsofthispeakefficiencyanalysismustbe correctedfortheadditionaldiffuserlosses.h general,however,thetrendstidi~tedby thisanalysiswillbe unaffected.Thisconstant-areaanalysisisnotapplicablefortunnelswithsecondthroats.

—..———— . —— .—

10 NACATN 2918 “

Considerationhasbeengiwn to theproblemoftestingan enginehavingan 8-inchoutletdiameterinan 18-by 18-tichtunneloperatingat a Machnumberof3.1. Tole foundareth effectsupontunneloperatingconditionscausedby variationsin z andtheexit-nozzlethroatareaoftheenginebeingtested.

~ thissolution,theenghe-airmass-flowratiowasassumedtobeunity.Equation(11)wasusedto ccmputevaluesofenginepressurerecoveryP2/Po forvariousvaluesofeat MachnumberM2 and %.In orderto stiplifytheproblem,itwasassumedthat A2/Ao= 1 andthattheenginewaEa perfectcylinder,makingtheccmditionsatstationsO and1 equal(seefig.4).

Withconditionsat stations1 and2 lamwn,theconditionsinthetunnelaftermixingwereobtainedby useofequations(9)and (4). Theresultsofthesecomputationsareplottedinfigures8 and9.

Figure8 showstunnelMachnumberaftermixingdownstreamofthenormalshockinthetestsectionM3, asa functionofenginepressurerecoveryforvarioustotaltemperatureratios.Linesof constantengine-exitMachnumberme shown.Figure9 showsresulttigtunnelpressure

4recoveryF3,/Pl as a functionof he sameindependentvariables.

Increasesintotal-temperatureratio,whileholdingeitherM2 orP2/Po constant,resulth highervaluesof M3, and P3t/P1.Anticreaseintheenginepressurerecovery,whileholdingthetotal-temperatureratioconstant,ticreasesthetunnelpressurerecoveryanddecreasesthevalueof M3t.

Theincreaseh thetunnelpressurerecoverieswithincreasingtotal-temperatureratios(greaterheataddition)isexplainableasfOllows. lh?cmthetheoryofOne-dimensionalgasflow(ref.1),thesefactsmaybe statedaboutconditionsat statim3’: (1)At a constantvalueoftotaltemperature,decreasesh totalmamentum(@(l + 7M2))atthisstationwillresultin ticreasedvaluesof M3, anddecreasedtunnelpressurerecoveryP3,/Pi,and (2)Iftotalmomentumisheldconstant,increasesinenergywillresulttngreatervaluesof M3tandsmallertunnelrecoveries.

#

lh figures8 and9,linesof constantz arelinesof constantenergy(constantT31). Linesof constanttotalmomentum .

(aX= constant)havebeendrawnon.ea$hfigure.Iargervaluesof~

NACATN 2918 11

.aX areaynonomouswithgreatertotalmomentumat statioq3’ (eq.(4)),as station1 representstunnelfree-streamconditionswhichareconstant.If changesareolservedalongtheseenergycmdmomentunlines,theresultsareseentobe inagreementwiththeone-dimensionalgas-flowtheory.Variationsin P31/P1withchangesin fuel-airratiosarethereforean effectof crxnbinedmomentumandenergychanges.

CONELUDINGRIMARKS

A one-dtiensicmalanalysisoftheresultsoftheparallel-Jetmixingencounteredinthetestingofenginesh supersonicw3ndtumnelshasbeenreported.Thistypeofanalysispresentsa reasonableapproachto obtainingapproxhatefiguresforthetunneloperattigconditionswhilethetunnelisstill3nthedesignstage.Thesefigureswouldbebasedupontheknowntunnelgecunetryandinletconditionsandesttiationsofthemodelgeometryandvaluesoftheengine-performanceparameters.Additionalequatiohsarepresentedforevaluationof changesdueto theburningofexcessfueldownstreamoftheengine-exhauststation.

Intheevaluationofthepropertiesoftheengine-exhaustjetandtheresultant(mixed)stream,ithasbeendemonstratedthattheassumptionsofconstantCP md 7 introduceno signffioanterrorsinthefinalresults.

LewisFlightPropulsionlaboratoryNationalAdvisoryCommitteeforAeronautics

Cleveland,Ohio,January5, 1953

I-2 NACATN 2918

AX’PENDIXA *

DISCUSSIOIVOl?ERRORINVOLVEDINASSDWTIONOF CONSTANT7 AND Cp

h viewofthevaluesoftemperatureassociatedwiththetestingofenginesh supersonicwindtunnelsandthevariationsof ~ and 7 at N+thesetemperatures,theassumptionbf constantCP and 7,asmadeh Pc1themixingsolutionandinthedetemhationof exhaust-~etpropetiles,appesxsjustified.

To illustrate,itisassumedthatthetotaltemperatureoftheexhaustjetislimitedto a maxhumof3000°R, a valuetasedapprox-imatelyonthehighesttemperatureswhichpresent-daymaterialscanwithstand. It isalsoassumedthattheetiust-jetMachnumberwillbegreaterthanorequalto 1.6,a reasonablevalueforan engineoperatingina supersonicstreamofmoderateMachnumber.Thesefactorsplaceanupperlimitofapproximately2000°R onthestatictemperatureoftheexhauststream,foran extremecase.It isto be realizedthatastheetiust-~etMachnumberticreasesortotaltemperatedecreasesorboth, “thevalueof itsstatictemperaturedecreases.

Ihtheusualsupersonicwindtunnel,airis expandedto a tempera-tureoftheorderof200°R, althoughthisfiguremaybe decreasedcon-siderablyforhypersonictunnels.Hence,therangeofstatictemperaturesofimportanceinthe~ problemisfrom200°to 2000°R. Themixedstream,priortothetunnelterminalshock,wouldhavea statictempera-turegreaterthan,butmuchcloserto,the200°R figure,If supersonicmixingexisted.Thisisa resultofthesmallmassflowthroughtheengine(hotair)as ccnnparedwiththatthroughthetunneladjacenttotheengtie.Thetunneltermhalshockmaycausethestatictemperatureto increaseby a factorof2 or3,buta valueof2000°R isstillfelttobe anupperlimitofstatictempe~tureina veryextremecase.

Forair,the~iations in Cp and 7 are (ref.2):

t,OR

200500100015002000 t

CP‘ 7Btulb-%0.23951.400.2400l.iOO.24881.380.26$41.349.27761.328

,1

.

NACATN 2918 13

For2000°R thevariationsin Cp and 7 fromthevaluesof200°R* are15.8and5.2percent,respectively,althoughthesemriationsdecrease

rapidlyfortemperatureslessthan2000°R. b therangeoftemperaturesfrom200°to 540°R,both Cp and 7 areconstant.

Nc)g To shuwtheeffectsupontheaccuracyoftheresultsofthemixing

problemwhen 7 and ~,areassumedconstant,thefol.lowingproblemwasconsidered:

Tunnelsize,18in.hy 18in.

Modelsize,8 in.diam.

M. = 3.1

M2 = 1.76

To= 550°R

T2 = 2370°R.

~c= 100percent

Theexhaust-jetpropertieswerefoundfromthesedata,andthenthemixingequationswereapplied,bothwith Cp and 7 assumedconstantandwithvaluesof Cp and 7 dependentuponthetemperature.Thelattermethodinvolveda trial-and-errorsolution,thedetailsofwhicharenotpresentedhere.A tabulationofresultsforbothcases,alongwithper~entagevariations,follows:

EConstant~endyVariableC$and 7

CP,2Variationh %’peroent

I

--l0.24009.6

0.263I

y2 Variation~, VwiationIn 7, h M31,peroent peroent

i.400 0.5803.4 2.6

1.352 0.565

P3,/P1

0.332C

0.328E

Variationin

percent

1.1,

Largevariationsin CP &d 7 aresbentore’suitinverysmallvaria-tionsinthefinalresults.

—— .— ————

14 NACATN 2918

AFPENDIXB “

BEATADDITIONINMIXINGREGION

Theequationgovemklngtheflowofa fluidin a constant-areachannelwithheatadditioncanbewritten

Momentumequatim

p(l+ mz) = p“(l+

Divisionofthecontinuityequationby

(Bl)

7M”2) (B2)

themomentumequationyields .

F’WR=[””=J(.,

Equation(B3),whensquared,becomes

()

l-y,,

Ny’ P “

Applicationofthisrelationtothesolutionfields

()T3,1

— =q311~3 T3

(B4)

ofthemixingproblem

(B5)

Thereforeit isseentkt Q3i’canbe directlyobtatiedfrom ~ by

theequation

Theratio !T3tl/T3isgiventitermsoffuelheattigvalueandenginecmnbustionefficiencyas

(B6)

NACATN 2918 15

r

‘~-s where

and

()1+=T3 =T1 3

1+%’Combiningequations(B7),(B8),and (B9)yields

+1

Thecombustionefficiencyqc isgiven

men Q311hasbeeneyaluated,.M3,,may

approxhnatelyas

be‘obtainedfrom

(B7)

(I!a)

(B9)

(B1O)

(Bll)

figure1. Choiceofthesubsonicor supersonicsolution”isdetez’minedby thesamefactorsdiscussedinrelationto equaticm(9)orsupersonic—windtunnels.

Theconservationofmoment~andmassflowprocessallowstheuseofe@atione(4)tid(6)and t31f,respectively,fmm thevalueof M31r...

-.

duringtheheatfordetermtiing

additidnp3n

A ‘compariscmofequations(9)and (B6)illustratestheeffectof.. heatadditioninthemtx@ regim. Since T31!/T3isalwaysgreater

thanunityforsucha heataddition,then 9311>Q3, andhence.

. .—— —— —.— — —_______ ———

16 NACATN 2918

M31!>M31.AdditionofheatinthemixingregionthereforecausesanincreasetithesubsonicMachnumberat statim3. A furthereffect,as indicatedinreference1,isto causean ticreaseh theentropyofthestream,witha subsequentdecreaseintotalpressureat station3.

1.

2.

Shapiro,AscherH.,andHawthorne,W. R.: TheMechanicsandThermo-dynamicsofSteadyOne-DtiensionalGasl?low.Jour.Appl.Mech.,vol.14,no.4,Dec.1947,pp.A317-A336.

Keenau,JosephH.,andKaye,Joseph:ThermodynamicPropertiesofAir. JohnWiley& Sons,Ihc.,1.945.

.

.

3R NilCATN 2918

Ie

17

10 9 8 ‘7 6 5 4 3 2 1Machnumber,M.,

Figure1. - Variationof functionq withMachnumberforal??(ratioof specificheats,1.4).

—.——- —.— —

I

.7?

.?(

.E

.&!

.55

.SC

.4.5

,4C

.ss.1

a1

Fig.um2. . Effect of rariat.im in ~ters of mldmg aquation on Mach mmkr after mixing (~ - S.0).

,

-+.a’

I““o~

.325E

52”L.s10r

—

w

—

—

—

—

2814, ,

1 1a

(a)Ratio of Kmh numbers of atreum being-d c equalto 1.

b

Effectof variation of pwwmaters in mixing equation on tolal-premsu.m ratio of ’rdxiq proooas‘r-%, P, - I+).

I I I I I I II I I II I I I I I I I I I I

10

(b) Ratio or Stntio tipermmm in stream being mixed 13 ml M 1. N

?W.m S. - C.molnM. Eff..tof .mrUticmofr.ummtaah mia ewntion cmtaal-wemttm ratio or mitigprwoem (H1. so; ~ . ~]. $co

1

2814 – “—_.__—--—— 6

EngineSupersonic Jet

—— — -—“~.

–T––’

<

&/

–L––2 2e~ -—. —

Figwe 4; - Schematic’ill~trat~on

Normal shock1

of engine being tetitedin

‘Wixingregion

supersonic

-,

wind tuhnel.

Nr

22 NACATN 2918

.

Figore5. - Photographaup=sonicstreamof

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